In short, Blockchain is a technology for keeping secure, ordered, decentralized public records of any transaction between any two parties in a verifiable way. Batches of such transactions are called blocks. They are added sequentially one after another with a digital fingerprint of the previous block, thus forming chains. Blockchain is secure by design. It keeps copies of its database across its network and requires all copies to agree to confirm new transactions. So no transaction information can be rewritten retroactively.
According to Wikipedia, "decentralised consensus can be achieved with a blockchain. This makes blockchains suitable for the recording of events, medical records, and other records management activities, identity management, transaction processing and proving provenance. This offers the potential of mass disintermediation and vast repercussions for how global trade is conducted."
The technology is already there. It's already being used to record Bitcoin and other cryptocurrency transactions. It's not going to stop at that. And, like mobile phones and the internet, it will gradually become an integral part of our lives. And it will affect not only how we get and spend money, but many other aspects of our lives, including, ultimately, how we interact, deal and trade with each other.
There are two excellent TED talks on the subject that will give you more interesting insights, and a couple of very detailed articles on Wikipedia and O'Reiley as well - wikipedia.org & Understanding the blockchain